Algebraic and group structure for bipartite three dimensional anisotropic Ising model on a non-local basis
AAlgebraic and group structure for bipartite three dimensional anisotropic Ising modelon a non-local basis
Francisco Delgado
1, 2, ∗ Escuela Nacional de Posgrado en Ciencias e Ingenier´ıa, Tecnol´ogico de Monterrey, M´exico. Departamento de F´ısica y Matem´aticas, Tecnol´ogico de Monterrey,Campus Estado de M´exico, Atizap´an, Estado de M´exico, CP. 52926, M´exico. (Dated: August 8, 2018)Entanglement is considered as a basic physical resource for modern quantum applications inQuantum Information and Quantum Computation theories. Interactions able to generate and sus-tain entanglement are subject to deep research in order to have understanding and control on it,based on specific physical systems. Atoms, ions or quantum dots are considered a key piece in quan-tum applications because is a basic piece of developments towards a scalable spin-based quantumcomputer through universal and basic quantum operations. Ising model is a type of interactionwhich generates and modifies entanglement properties of quantum systems based on matter. In thiswork, a general anisotropic three dimensional Ising model including an inhomogeneous magneticfield is analyzed to obtain their evolution and then, their algebraic properties which are controlledthrough a set of physical parameters. Evolution denote remarkable group properties when is ana-lyzed in a non local basis, in particular those related with entanglement. These properties give afruitful arena for further quantum applications and their control.
PACS numbers: 03.67.-a; 03.67.Bg; 03.65.Ud; 03.65.Ge; 03.65.Fd; 03.65.Aa; 02.20.Uw
I. INTRODUCTION
Quantum entanglement is one of the most interestingproperties of Quantum Mechanics which was noted sinceearly times of theory [1–5]. Nowadays, this property hasbeen exploited by quantum applications as central aspectto improve information processing in terms of capacityand speed [6–8]. Thus, Quantum Information studies en-tanglement as an important aspect to codify and manageinformation in several quantum applications developedsince seminal proposals in Quantum Computation [9–11],Quantum Cryptography [12, 13] and discoveries aboutsuperdense coding [14] and teleportation [15]. A com-plete entanglement map of road will not be constructeduntil its quantification and behavior could be understoodsince a general mathematical theory and a deep knowl-edge about quantum interactions which generates it. Itlast means, Hamiltonian models which are able to gen-erate entanglement, which are actually studied in orderto understand how this quantum feature is generated onseveral physical systems. For magnetic systems, Isingmodel [16, 17] in statistical physics and Heisenberg model[18] in quantum mechanics are Hamiltonian models de-rived from interaction between spin systems when theyinclude a magnetic field, it works as a driven element inHamiltonian. Nielsen [19] was the first reporting studiesof entanglement in magnetic systems based on a two spinsystems using that model including an external magneticfield.Magnetic driven Ising interaction is well known bydeveloping an evolution depending on local parameters. ∗ Electronic address: [email protected]
Still its simplicity, for only two particles it exhibit fourenergy levels introducing a non periodical behavior interms of Rabi frequencies phenomenon and their control[20, 21]. Several simplified models has been analyzed inorder to understand quantum behavior of these kind ofsystems when they approach to different concrete sys-tems as quantum dots or electronic gases. Still, researcharound of control and entanglement in bipartite qubits[22] and lattices [23, 24] is fundamental because thesesimple systems let the possibility to control quantumstates of a single or a couple of electron spins at time,standing at the heart of developments towards a scalablespin-based quantum computer.Control being depicted, in combination with controlledexchange between neighboring spins, would let obtainuniversal quantum operations [25–27] in agreement withDiVincenzo criteria [28] in terms of reliability of statepreparation and identification of well identified qubits.Thus, the aim of this paper is analyze algebraic proper-ties of a bipartite system with a general three dimensionalanisotropic Ising interaction including an inhomogeneousmagnetic field strength in a fixed direction. One of thecentral aspects is that analysis of dynamics is conductedon a non-local basis in terms of classical Bell states, whichlets to discover outstanding algebraic aspects of this in-teraction around entanglement and a regular group struc-ture, obtaining possible direct applications for quantumcontrol and quantum computer processing.
II. ANISOTROPIC ISING MODEL IN THREEDIMENSIONS
Different models of Ising interaction (XX, XY, XYZdepending on focus given by each author) has been con- a r X i v : . [ qu a n t - ph ] O c t sidered in order to reproduce calculations related withbipartite and tripartite systems [29–31]). Similar mod-els which requires interaction with radiation are modeledin terms of Jaynes-Cummings and Jaynes-Cummings-Hubbard Hamiltonians [32–34]. As example, in quantumcontrol, different versions of Ising interaction have beenconsidered in terms of homogeneity of magnetic field, di-mensions and directions involved [35–38]. Thus, restric-tions in dimensions, number of particles and strength ofexternal fields in these models are due for simplicity, ge-ometry of lattices and other properties of physical sys-tems involved [30, 39–42].In this work, we focus on the following Hamiltonianfor the bipartite anisotropic Ising model [16, 19] includ-ing an inhomogeneous magnetic field restricted to the h -direction ( h = 1 , , x, y, z respec-tively): H h = − σ · J · σ + B · σ + B · σ = − (cid:88) k =1 J k σ k σ k + B h σ h + B h σ h (1)which attempts to generalize most of several models con-sidered in the cited works before. By diagonalizing andfinding the corresponding eigenvalues, which are inde-pendent of h = 1 , , E h (1) = − J h − R h + , E h (2) = − J h + R h + (2) E h (3) = J h − R h − , E h (4) = J h + R h − where R h − and R h + are defined as follows: if h, i, j is acyclic permutation of 1 , , { h } symbol, being equivalent to the pair of indexes i, j : R h ± = (cid:113) B h ± + J i,j ∓ = (cid:113) B h ± + J { h }∓ (3)with : J { h }± ≡ J i,j ± = J i ± J j B h ± = B h ± B h A. Reduced notation and definitions
One more suitable selection of reduced parameters willestablish an appropriate notation which we will follow inthe whole of work (in order to make finite some parame-ters and to reduce extent of some expressions): b h ± = B h ± R h ± , j h ± = J { h }∓ R h ± ∈ [ − ,
1] (4)Note that subscripts − , + are settled for these vari-ables in relation with their internal operations in (4). Itis known that when anisotropic Ising evolution matrix is expressed in the computational basis, it has in general acomplex full form (with full 4 × z direction[37, 38], which denotes the privileged basis selected. Forthis reason, in the next section, the analysis will be basedon Bell state basis to develop a different structure in it.This selection suggests to change the notation by using − , + for lower and upper scripts. Nevertheless, whenthese labels appear in mathematical expressions, it willbe convenient recover them to express operations, so theywill be assumed as − , +1 respectively when they areforming part of algebraic expressions. Reader should bealert about it. In the same sense, capital scripts A, B, ... will be reserved for 0 , − , +1 or − , +; latinscripts h, i, j, k, ... will reserved spatial coordinates x, y, z or 1 , ,
3; and a, b, c, ... (between parenthesis) as scriptsfor energy levels 1 , , , · will be used sometimes to em-phasize number multiplication between terms in scriptsand avoid confusions. For example, in this notation wewill write the standard Bell states as: | β −− (cid:105) ≡ | β (cid:105) , | β − + (cid:105) ≡ | β (cid:105) (5) | β + − (cid:105) ≡ | β (cid:105) , | β ++ (cid:105) ≡ | β (cid:105) B. Eigenvectors and Evolution operator
In last terms, E ( a ) h : E (1) h , E (2) h , E (3) h , E (4) h correspondwith E µν : E −− , E − + , E + − , E ++ respectively and theycould written in the current notation as: E hµν ≡ E h (2+ µ + ν ) = µJ h + νR h − µ (6)= µJ h + ν (cid:113) B h − µ + J { h } µ the corresponding eigenvectors for each direction h are: (cid:12)(cid:12) φ µν (cid:11) = (cid:88) (cid:15) ∈{− , + } δ + (cid:15) ν (1 + µνj − µ ) − δ − (cid:15) µb − µ √ (cid:112) νµj − µ | β µ(cid:15) (cid:105) (cid:12)(cid:12) φ µν (cid:11) = (cid:88) (cid:15) ∈{− , + } δ + (cid:15) iν (1 + µνj − µ ) + δ − (cid:15) µ(cid:15)b − µ √ (cid:112) νµj − µ | β µ · (cid:15)(cid:15) (cid:105) (cid:12)(cid:12) φ µν (cid:11) = (cid:88) (cid:15) ∈{− , + } (1 + νb − µ ) + ν(cid:15)j − µ (cid:112) νb − µ | β (cid:15)µ (cid:105) (7)where δ αβ is the custom Kronecker delta.An arbitrary bipartite state is written in computa-tional basis or in Bell basis respectively as: | ψ (cid:105) = (cid:88) A,B ∈{ , } A AB | AB (cid:105) = (cid:88) α,β ∈{− , + } B α,β | β αβ (cid:105) (8)then it is possible demonstrate by direct calculation thatconcurrence and Schmidt coefficients [43], in terms of coe-fficients B α,β , are in the Bell basis: C = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) αβ ∈{− , + } β B α,α · β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (9) λ | ψ (cid:105) = 12 (cid:16) ± (cid:112) − C (cid:17) (10)so, for eigenstates (cid:12)(cid:12) φ iµν (cid:11) , coefficients are simply: λ | φ hµν (cid:105) = 12 (1 ± (cid:12)(cid:12) b h − µ (cid:12)(cid:12) ) (11)then, their entropy of entanglement becomes: S | φ hµν (cid:105) = 1 − (cid:88) ν ∈{− , + } (1 + ν (cid:12)(cid:12) b h − µ (cid:12)(cid:12) ) log (1 + ν (cid:12)(cid:12) b h − µ (cid:12)(cid:12) )(12)which is maximal only if b hµ = 0 (symmetric or antisym-metric fields), in agreement with [37]. III. FORM AND STRUCTURE OF EVOLUTIONOPERATOR
Using last expressions for eigenvalues and eigenvectors,and introducing the following convenient definitions re-lated with energy levels:∆ hνµ = t E hµ + + νE hµ − ) = (cid:40) µJ h t if ν = + R h − µ t if ν = − (13) and: e hβα = cos ∆ h − α + iβj h − α sin ∆ h − α (14) d hα = b h − α sin ∆ h − α then, if evolution operator is written in Bell basis as: U h ( t ) = (cid:88) α,β,γ,δ U hαβ,γδ | β αβ (cid:105) (cid:104) β γδ | (15)so, for those elements different from zero, its explicit formbecomes : U αβ,µν = δ αµ e i ∆ α ( δ βν e α · βα − iα (1 − δ βν ) d α ) (16) U αβ,µ · αν · β = δ µν e i ∆ α · β ( δ +1 µ e αα · β + δ − µ αd α · β ) U αβ,µν = δ βν e i ∆ β ( δ αµ e αβ + i (1 − δ αµ ) d β )which gives close forms for evolution operators in eachcase when they are expressed in the non-local basis ofBell states. It express, in some sense, more explicitlythe evolution of entanglement. Reader should note thatscripts − , + in variables defined in current section arerelated with energy labels more than internal operations(as in J { h }± , B { h }∓ ) as before. Awareness about thisdetail in notation will avoid later misconceptions. A. Sector structure in the Evolution operator
Last expressions could be appreciated better in matrixform: U ( t ) = e i ∆ − e −−∗ ie i ∆ − d − ie i ∆ − d − e i ∆ − e −− e i ∆ e ∗ − ie i ∆ d − ie i ∆ d e i ∆ e ∈ S (17) U ( t ) = e i ∆ e ∗ − e i ∆ d e i ∆ − e −∗ − e i ∆ − d − e i ∆ − d − e i ∆ − e − e i ∆ d e i ∆ e ∈ S U ( t ) = e i ∆ − e −∗ ie i ∆ − d − e i ∆ e ∗ ie i ∆ d ie i ∆ − d − e i ∆ − e − ie i ∆ d e i ∆ e ∈ S U h =1 , , ( t ) clearly have a 2 × I is included by setsimply t = 0. By the properties of e hβα and d hα , sectorsare unitary with e i ∆ h + α as determinant. Because U h ( t )is unitary, inverses are obtained just by take U † h ( t ) (nev-ertheless those structure, it is required prove if it can beobtained as U h ( t (cid:48) ) for some other physical parameters forthe same system). In addition, as the sum of eigenvaluesis zero, then U ( t ) ∈ SU (4), which is an important aspectof our evolution operator.Referring only to their structure, 4 × SU (4) formed by unitary 2 × U (2)as is depicted in (17) (clearly with unitary and reciprocaldeterminants), they form groups. Thus, it is easy showthat each one, S , S , S , are subgroups of SU (4) (iden-tity, inverses and multiplication are included in each S h and product of two elements in the set remains in it): S = { A ∈ SU (4) | A αβ,γδ = δ αγ u αβ,γδ , ( u γα,γβ ) γ = ± ∈ U (2) } (18) S = { A ∈ SU (4) | A αβ,γδ = δ α · γβ · δ u αβ,γδ , ( u αβ,γ · αγ · β ) γ = ± ∈ U (2) } (19) S = { A ∈ SU (4) | A αβ,γδ = δ βδ u αγ,βγ , ( u γα,γβ ) γ = ± ∈ U (2) } (20)where α, β, γ, δ ∈ {− , + } . In addition, we state the sym-bol S ∗ h ⊂ S h to each set of matrices able to be generatedby U h ( t ) in (17). In them, the general structure for their2 × U (2) is: s hj = e i ∆ h + α (cid:32) e hβα ∗ − qi h d hα qi ∗ h d hα e hβα (cid:33) α = ( − h + j +1 β = ( − j ( h + lj − kj +1) q = β ( − h +1 (21)where h denotes the associated spatial coordinate of mag-netic field, j = 1 , k j , l j , the labels for its rows in each matrix of (17)(by example, k = 3 , l = 4 are the labels for the secondsector, j = 2, in U h =1 ( t ) it means s . Note particularlythat determinant for each sector, det( s hj ) = e i ∆ h + α , arereciprocal because ∆ h + − α = − ∆ h + α . Note that s hj ∈ U (2)as was previously stated, but not all elements of U (2)is a s hj for any parameters e hβα , d hα (because the formof − qi h d hα , qi ∗ h d hα in entries 1 , , π ). This implies that s hj is not necessarily a subgroup of U (2), which open opportunities to extend their coveragein U (2) with two o more pulses. For a further discussion, it is notable to write thegeneric sector in exponential form in terms of Pauli ma-trices: s hj = e i ∆ h + α e − i ∆ h − α n · σ ≡ e i ∆ h + α s hj, = e i ∆ h + α (cid:0) cos ∆ h − α I − i sin ∆ h − α n · σ (cid:1) (22)with : n = ( qb h − α sin hπ , qb h − α cos hπ , βj h − α )where σ = ( σ , σ , σ ), I n is the n × n identity matrixand s hj, ∈ SU (2) is the matrix sector with lacking of itsnumeric exponential factor, which is defined by furtherconvenience. That structure shows that each kind of in-teraction (with external magnetic field in h − coordinate)exclusively transforms Bell states in specific pairs as a SU (2) operation plus a phase term in U (1). Then, eachsector for a given h is responsible to combine linearlytwo Bell states | β µν (cid:105) and | β µ (cid:48) ν (cid:48) (cid:105) under a U (1) × SU (2)operation, with:( µ (cid:48) , ν (cid:48) ) = ( µ, ν ⊗ , h = 1( µ ⊕ , ν ⊕ , h = 2( µ ⊕ , ν ) , h = 3 (23)where ⊕ is the sum module 2. Thus, it is possible asso-ciate, by eliminating the sector phase e i ∆ h + α ∈ U (1) andinstantaneous exponential factor in component | β µν (cid:105) (itis equivalent to use a certain rotating frame to whole sys-tem), a Bloch sphere between | β µν (cid:105) and | β µ (cid:48) ν (cid:48) (cid:105) in which,each part of | ψ (cid:105) in (8) corresponding with each sector: | ψ j (cid:105) = (cid:88) γ,δ ∈{ µν,µ (cid:48) ν (cid:48) } B γδ | β γδ (cid:105) (24) ⇒ | ψ (cid:105) = (cid:88) j =1 , | ψ j (cid:105) evolves ’locally’ driven by s hj in sector j . Note that thisevolution does not change probabilities between partsin each sector, but introduces relative phases because e i ∆ h + α ∈ U (1) and the complex mixing generated by s hj, .Clearly s hj, are elements of a Lie group with parameters∆ h − α n , which will be important further.Figure 1 shows Bell states related with several inter-actions U x ( t ) , U y ( t ) , U z ( t ) or U ( t ) , U ( t ) , U ( t ) in (17).Each dotted line is a linear combination of states in theirvertex (24). Given a initial state of this type and depen-ding on interaction being considered ( h = 1 , , | β µν (cid:105) and | β µ (cid:48) ν (cid:48) (cid:105) related by an specific interaction in their re-spective Bloch sphere.By combining several adequate interactions, could bepossible switch any Bell state into another (Figure 1) ifsectors s hj adopt the following combined forms for t = T (referred in the following as diagonalization and antidi-agonalization cases respectively): s hj = ± I (25) s hj = ± σ or : s hj = ± iσ (26)Thus, by combining these two types of sector forms,we can achieve evolution loops [44–47] and exchange op-erations [38] in H ⊗ for Bell states, which will let obtainseveral control effects. Note that last expressions giveonly the matrix form for one sector s hj , avoiding a con-fusion between operators in computational basis and Bellbasis by the use of Pauli matrices σ , σ in last expres-sion, which are used only to set a desired form in thatmatrix sector. Clearly, much more equivalent cases couldbe considered with arbitrary phases instead of only ± B. Evolution of Bell states entanglement
In spite of (9) and (17), concurrence of states evolvedfrom Bell states could be easily obtained. Short calcula-
FIG. 1: Schematic representation of states related betweeneach interaction in (17). Each dotted line should be under-stood as linear combination of states on vertex. Each linerepresents an space equivalent to a Bloch sphere for relatedstates | β µν (cid:105) and | β µ (cid:48) ν (cid:48) (cid:105) . On this sphere, evolution of linearcombination could be represented as lines depicted in the pic-ture on the right driven by an operator in SU (2). tions show that concurrence C hµν for evolution U h ( t ) forBell state | β µν (cid:105) is easily expressed as: C hµν = 1 − j h − f hµν b h − f hµν sin ∆ − h f hµν (27)with : f hµν = µ , h = 1 µν , h = 2 ν , h = 3showing that it depends only on one Rabi frequencyat time in a very simply way. This expression is con-sistent with isotropic case reported in [37, 38]. Notethat (27) reproduces those results because this expres-sion does not imply that states comeback each period tothe original Bell state, only to the same entanglementvalue. Clearly, Bell states could become separable inter-mediately if j h f hµν b h f hµν = 1 /
4, which is possible if C hµν reaches its maximum value when | B h − f hµν | = | J { h } f hµν | .In isotropic cases, some Bell states are invariant whenmagnetic field is symmetric or antisymmetric, as wasshown in [38], which in general does not happen here.All this scenario contrasts with evolution of some separa-ble states, by example those of computational basis | ij (cid:105) ,whose entanglement expressions depends normally on allor several Rabi frequencies involved, generally resultingin a non periodical behavior. Thus, in some sense, en-tanglement instead of separability appears as a naturalfeature of Ising model inclusive with external magneticfields. C. Equivalence under rotations
Of course, last evolution operators are related via anhomogeneous bipartite rotation in terms of Euler angles[48, 49] on Hilbert space H and on Fock space H ⊗ : R , ( α, β, γ ) = R ( α, β, γ ) ⊗ R ( α, β, γ ) (28)where: R i ( α, β, γ ) = cos β α + γ I (29) − i sin β α − γ σ − i sin β α − γ σ − i cos β α + γ σ = (cid:32) e − i α + γ cos β − e i α − γ sin β e − i α − γ sin β e i α + γ cos β (cid:33) expressed in the computational basis. As is expected,different Ising models with magnetic fields in cartesiandirections transform between them. Specifically: U ( t ) = R , ( π , π , U ( t ) R , † ( π , π ,
0) (30) U ( t ) = R , ( − π, π , π U ( t ) R , † ( − π, π , π | (cid:105) and | (cid:105) . IV. GROUP STRUCTURE OF EVOLUTIONOPERATORS
Previously has been stated that evolution operators U h ( t ) are part of subgroup S h in SU (4) defined by theirform in (18). But there are a inner structure which canbe found in terms of group properties, which are notonly related with the form of these operators but insteadwith their quantum structure related with Ising Hamil-tonian (1). In this section, we analyze specific structureand restrictions in (17) to S ∗ h ⊂ S h ⊂ SU (4) becomes asubgroup, together with traditional operator or matrixproduct in terms of their physical properties. It means,the physical prescriptions on t, j h ± α , b h ± α parameters foreach sector with which U h ( t ) fulfills a group structure:Closure: U h ( t (cid:48) ) U h ( t ) = U h ( t (cid:48)(cid:48) ) ∈ S ∗ h ∀ U h ( t ) , U h ( t (cid:48) ) ∈ S ∗ h Associativity: U h ( t (cid:48)(cid:48) ) ( U h ( t (cid:48) ) U h ( t )) = ( U h ( t (cid:48)(cid:48) ) U h ( t (cid:48) )) U h ( t ) ∀ U h ( t ) , U h ( t (cid:48) ) , U h ( t (cid:48)(cid:48) ) ∈ S ∗ h Identity: I ∈ S ∗ h Inverse: U − h ( t ) = U h ( t (cid:48) ) ∈ S ∗ h ∀ U h ( t ) ∈ S ∗ h where U h ( t ) is understood to have the structure in (17).Clearly associativity and existence of identity (∆ h + ± α =2 mπ, ∆ h −± α = 2 nπ, m, n ∈ Z ) are fulfilled because cov-ering SU (4) structure. Thus, only the product closure and existence of inverse should be analyzed. Because S ∗ h ⊂ S h , sector structure is accomplished and analysisof previous conditions can be almost restricted to sectors s hj (note only that exponential factor in each sector isthe inverse of its respective factor in other sector, so bothshould be compatible in addition). A. Inverse
In spite of sector properties of matrices in S ∗ h , inverseof U h ( t ) reduces to obtain inverse of each sector (caringcompatibility around of their exponential factors). Be-cause generic sector (21) is unitary, its inverse is: s h − j = e − i ∆ h + α (cid:32) e hβα qi h d hα − qi ∗ h d hα e hβα ∗ (cid:33) (31)thus, conditions for a generic sector (21) mimicking lastexpression can be obtained by comparing entries 1 , , , , s (cid:48) hj = s h − j . This comparisonshows that there are two possible restrictions to makecompatible those four equations:∆ (cid:48) h + α + ∆ h + α = p α π, p α ∈ Z (32)or : e βα e (cid:48) αβ = e βα ∗ e (cid:48) αβ ∗ (33)with which, only two additional equations remains, byexample for entries 1 , , , e i (∆ h + α +∆ (cid:48) h + α ) e (cid:48) hβα ∗ = e hβα , − e i (∆ h + α +∆ (cid:48) h + α ) d (cid:48) hα = d hα (34)Equation (32) automatically fulfills the compatibilitybetween sectors because exponential factor becomes real.Combining this condition (34), we get several cases. Themost general is obtained noting that in spite of (34), b (cid:48) h − α = P α b h − α . After, to fulfill (34), it is required that∆ (cid:48) h − α + S α ∆ h − α = n α π which implies j (cid:48) h − α = S α j h − α , P α S α = 1 and p α = 2 n α + m α . With these conditions, s (cid:48) hα , with form (21), converts into (31). While, a briefanalysis of equation (33) into (34), shows that it reducesto a special case of last solution. Thus, the general pre-scriptions to get the inverse effect of an Ising interactionwith a similar interaction but changing physical param-eters are: j (cid:48) h − α = S α j h − α b (cid:48) h − α = S α b h − α ∆ (cid:48) h + α + ∆ h + α = (2 n α + m α ) π ∆ (cid:48) h − α + S α ∆ h − α = m α π (35)with : S α = ± , n α , m α ∈ Z Note that last prescriptions are compatible with pre-scriptions of evolution loops in two pulses. Still, thereis a pair of particular solutions. The first case, where P α S α = ± (cid:48) h + α + ∆ h + α = m α π, ∆ h − α = r α π, ∆ (cid:48) h − α = (2 n α + r α + m α ) π with n α , m α , r α ∈ Z which is the form of sectors in diagonalform D ∆ h + α h , whose inverse is simply D − ∆ h + α h : U ( T ) ≡ D φ = S , e iφ S , e iφ S , e − iφ
00 0 0 S , e − iφ U ( T ) ≡ D φ = S , e iφ S , e − iφ S , e − iφ
00 0 0 S , e iφ U ( T ) ≡ D φ = S , e iφ S , e − iφ S , e iφ
00 0 0 S , e − iφ (36)where S h,j are ± P α Sα , these prescriptions agree with (35).An possible additional case, which appears when equa-tions (34) are solved, is the case with ∆ (cid:48) h + α + ∆ h + α = m α π, ∆ h − α = r α +12 π, ∆ (cid:48) h − α = r (cid:48) α +12 π, P α S α = 1 , r α + r (cid:48) α + m α + 1 = S α . Nevertheless that this case is com-pletely included in solution (35), it states a specific kindof evolution matrices with s hj of form: s hj = ( − r α e i ∆ h + α (cid:32) − iβj h − α − qi h b h − α qi ∗ h b h − α iβj h − α (cid:33) (37) s h − j = ( − r (cid:48) α e i ∆ (cid:48) h + α (cid:32) iβj (cid:48) h − α qi h b (cid:48) h − α − qi ∗ h b (cid:48) h − α − iβj (cid:48) h − α (cid:33) (38)Note that when | j h − α | = | b h − α | = this sector remem-bers Hadamard-like gates.Nevertheless that prescriptions to get inverses are welldefined, they involves sometimes conditions on J hα , J (cid:48) hα which are not possible fulfill in specific experimental de-signs in terms to find t (cid:48) , B (cid:48) h − α in terms of t, B h − α . In-stead, sometimes some of J h , J (cid:48) h , J { h }± , J { h }± should ful-fill strong restrictions (precisely those equivalent to Evo-lution Loops prescriptions stated before). Last meansthat inverse evolution for one pulse is not always achiev-able under general restrictions with other one pulse evolu-tion, despite of inverse is well defined as part of matrices(17). B. Product closure
To probe the product closure and to obtain the pre-scriptions on physical parameters, we can restrict ourdiscussion to sectors again, caring the global matchingbetween exponential factors in (21). Thus, it is neces-sary probe the matching between s (cid:48) hj s hj in: s (cid:48) hj s hj = e i (∆ (cid:48) h + α +∆ h + α ) × (cid:32) e (cid:48) hβα ∗ e hβα ∗ − d (cid:48) hα d hα − qi h ( e (cid:48) hβα ∗ d hα + e hβα d (cid:48) hα ) qi ∗ h ( e (cid:48) hβα d hα + e hβα ∗ d (cid:48) hα ) e (cid:48) hβα e hβα − d (cid:48) hα d hα (cid:33) (39)with s (cid:48)(cid:48) hj with the form in (21). Demonstration is straightbut it requires some transformations which we outlinebriefly. Again, comparing sectors we note that respectiveequations are compatible only if:∆ (cid:48)(cid:48) h + α = ∆ (cid:48) h + α + ∆ h + α + 2 r α π (40)with n α ∈ Z . If this equation fulfills, then only twoequations remain, for entries: 2 , e h − βα d (cid:48) hα + e (cid:48) h − βα d hα = d (cid:48)(cid:48) hα (41)2 , e (cid:48) hβα e hβα − d (cid:48) hα d hα = e (cid:48)(cid:48) hβα (42)Equation (42) traduces in two equations for real andimaginary parts respectively:cos ∆ (cid:48)(cid:48) h − α = cos ∆ (cid:48) h − α cos ∆ h − α − ( j (cid:48) h − α j h − α + b (cid:48) h − α b h − α ) · sin ∆ (cid:48) h − α sin ∆ h − α (43) j (cid:48)(cid:48) h − α sin ∆ (cid:48)(cid:48) h − α = j (cid:48) h − α sin ∆ (cid:48) h − α cos ∆ h − α + j h − α cos ∆ (cid:48) h − α sin ∆ h − α (44)then, applying the following transformations δ ± = ∆ (cid:48) h − α ± ∆ h − α , δj ± = ( j h − α ± j (cid:48) h − α ) , δb ± = ( b h − α ± b (cid:48) h − α ), lastequations become:cos ∆ (cid:48)(cid:48) h − α = ( δj + δb ) cos δ + +( δj − + δb − ) cos δ − (45) j (cid:48)(cid:48) h − α sin ∆ (cid:48)(cid:48) h − α = δj + sin δ + + δj − sin δ − (46)Following the same process for equation (41), we obtainthat equations for real and imaginary parts become: b (cid:48)(cid:48) h − α sin ∆ (cid:48)(cid:48) h − α = δb + sin δ + + δb − sin δ − (47)0 = ( δj + δb − − δj − δb + )(cos δ + − cos δ − )= ( b h − α j (cid:48) h − α − b (cid:48) h − α j h − α ) · (cos δ + − cos δ − ) (48)Still, we require that right side in each equation (45-48)really represent the expression on their left side. Moreprecisely, we need that the following expression becomesone: cos ∆ (cid:48)(cid:48) h − α + ( j (cid:48)(cid:48) h α + b (cid:48)(cid:48) h α ) sin ∆ (cid:48)(cid:48) h − α == 1 − ( δj + δb )( δj − + δb − )(cos δ + − cos δ − ) = 1 − − ( j h − α j (cid:48) h − α + b h − α b (cid:48) h − α ) · (cos δ + − cos δ − ) (49)which, together with (48) should be fulfilled. There aretwo possible solutions: a) δ + = ± δ − + 2 nπ, n ∈ Z , whichis equivalent to ∆ h − α = nπ or ∆ (cid:48) h − α = nπ , leaving oneof them free. This solution is trivial because it impliesthat one of matrices in the product is proportional tothe identity; b) j (cid:48) h − α = S α j h − α , b (cid:48) h − α = S α b h − α , whichautomatically implies | S α | = 1. This situation is trivialwhen S α = 1 and J (cid:48){ h } α , J { h } α do not change because itmeans constant fields. But if interaction strengths couldbe manipulated, then it means (for S α = ± J (cid:48){ h } α J { h } α = B (cid:48) h − α B h − α (50) There is one relevant aspect which could be noticedhere departing from expressions (45-48). By consideringthat all parameters j h − α , j (cid:48) h − α , b h − α , b (cid:48) h − α , ∆ h − α , ∆ (cid:48) h − α are free (until their internal and relative natural restric-tions), these right side expressions have a variation be-tween [ − ,
1] in such way that magnitude of each entrycan reach the unit value as before. Nevertheless, by com-parison with one pulse case where only multiple integeror multiple semi-integer values are possible, phases in an-tidiagonal entries can, independently from diagonal ones,reach values in a continuous range inside of [0 , π ) butwithout cover last complete range. This behavior suggestthat multiple pulses could extend s hj coverage into U (2)(or easier, s hj, coverage into SU (2)).Summarizing, the prescriptions for product closure inthe last terms are: j (cid:48)(cid:48) h − α = j h − α = S α j (cid:48) h − α b (cid:48)(cid:48) h − α = b h − α = S α b (cid:48) h − α ∆ (cid:48)(cid:48) h + α = ∆ (cid:48) h + α + ∆ h + α + 2 r α π ∆ (cid:48)(cid:48) h − α = ∆ (cid:48) h − α + S α ∆ h − α + 2 r (cid:48) α π (51)with : S α = ± , r α , r (cid:48) α ∈ Z showing that each family of evolution matrices (17) donot form a group at least that j (cid:48) h − α = S α j h − α , b (cid:48) h − α = S α b h − α (which automatically implies | S α | = 1. This as-pect is important because it implies that effect of twoo more pulses of magnetic field can not be always re-placed with effects achievable by one pulse. These re-sults are consistent with Baker-Campbell-Hausdorff for-mula for SU (2) reported in [50]: s hj s h (cid:48) j = e i (∆ h + α +∆ (cid:48) h + α ) (cid:16) (cos ∆ h − α cos ∆ (cid:48) h − α − sin ∆ h − α sin ∆ (cid:48) h − α n · n (cid:48) ) I − i (sin ∆ h − α cos ∆ (cid:48) h − α n + cos ∆ h − α sin ∆ (cid:48) h − α n (cid:48) + sin ∆ h − α sin ∆ (cid:48) h − α n × n (cid:48) ) · σ (cid:17) ≡ e i ∆ (cid:48)(cid:48) h + α (cid:16) cos ∆ (cid:48)(cid:48) h − α I − i sin ∆ (cid:48)(cid:48) h − α n (cid:48)(cid:48) · σ (cid:17) (52)with : n = ( qb h − α sin hπ , qb h − α cos hπ , βj h − α ) n (cid:48) = ( qb (cid:48) h − α sin hπ , qb (cid:48) h − α cos hπ , βj (cid:48) h − α )where × denotes the vector product. Then, clearlyif n (cid:48) = S α n , | S α | = 1, both vectors become paralleland prescriptions (51) are recovered by comparing with(22). Note that U (1) factor is abelian and only the part s (cid:48)(cid:48) hj , ∈ SU (2) exhibit a more complex form. Particu-larly a non vanishing term sin ∆ h − α sin ∆ (cid:48) h − α n × n (cid:48) makesnon commutative both sectors.Coverage of two o more pulses s hj, into SU (2) couldbe understood from expression (52). Note that cos ∆ (cid:48)(cid:48) h − α ranges in [ − ,
1] independently of n (cid:48) · n = ( b h − α b (cid:48) h − α + j h − α j (cid:48) h − α ), when ∆ h − α = nπ or ∆ (cid:48) h − α = nπ with n ∈ Z ,leaving other parameter free. Last is required to s hj have a complete coverage into SU (2). Unfortunatelywhile this variation is nearer from ± | n · n (cid:48) | = 1,∆ h − α = nπ or ∆ (cid:48) h − α = nπ ), n (cid:48)(cid:48) have a more limited cover-age on unitary sphere as is required to reproduce SU (2). C. Group structure
In spite of formulas (35) and (51), it is clear that S α can be absorbed as a coefficient of ∆ h − α . Otherwise,last is equivalent to fix j h − α positive in each group el-ement together with its sign of b h − α . Because (22), lastis equivalent to group those elements by a common uni-tary vector n = ( qb h − α sin hπ , qb h − α cos hπ , βj h − α ). In S h ⊂ SU (4), each group is characterized by a set of twofixed values | j h ∓ α | and two signs s b h ∓ α correspondinglywith signs of b h ∓ α . Thus, each group can be labeled by {| j h ∓ α |} , { s b h ∓ α } and we will denote it by S ∗ h {| j h ∓ α |}{ s bh ∓ α } (be-ing ± α , the first and second sectors in U h ( t ) respectively,or j = 1 , C h ≡ (cid:91) | j h ± α |∈ [0 , s bh ∓ α = ± S ∗ h {| j h ∓ α |}{ s bh ∓ α } ⊂ S h ⊂ SU (4) (53)Clearly C h = { U h ( t ) } contains all evolution matri-ces generated by each U h ( t ) in (17) just grouped by | j h ∓ α | and s b h ∓ α for each sector, which precisely formsubgroups in S h . Note that parameters ∆ h ±± α generatemultifolded group elements because periodicity of expres-sions, it means, several of them generate same group ele-ments in each S h but they correspond to different evolu-tion dynamics. Same is true for terms 2 r α π, r (cid:48) α π in (51),which state several alternatives to reproduce an equiva-lent evolution matrix for two consecutive pulses, morethan r (cid:48) α , r α = 0. Unfortunately for control purposes, thisperiodicity is not always in the time variable. Thus, interms of last subsection, each subgroup S ∗ h {| j h ∓ α |}{ s bh ∓ α } (foreach h = 1 , ,
3) becomes an abelian group. In addition,because that ∆ h + α = − ∆ h + − α is not a global phase in U h ( t ) at least that it reduces to ±
1. Then these relativephase works as an interference frequency in superpositionstates containing terms of Bell states related with bothsectors (separable or partially entangled).Because rotations explained before, R , ( α, β, γ ) ∈ SU (4), transforms elements of each S h into other S h (cid:48) ,then subgroups S ∗ {| j ± |}{ s b ± } , S ∗ {| j ∓ |}{ s b ∓ } , S ∗ {| j ± |}{ s b ± } are clearlyisomorphic [51] between them under these rotations.Moreover, if T is a transformation with properties: T : s hj ∈ S ∗ h {| j h ∓ α |}{ s bh ∓ α } → s thj ∈ S ∗ h {| j (cid:48) h ∓ α |}{ s b (cid:48) h ∓ α } T ( s (cid:48) hj s hj ) = T ( s (cid:48) hj ) T ( s hj ) (54)then, in spite of (51), S ∗ h {| j h ∓ α |}{ s bh ∓ α } and S ∗ h {| j (cid:48) h ∓ α |}{ s b (cid:48) h ∓ α } becomeisomorphic because product structure is preserved: s (cid:48)(cid:48) thj = T ( s (cid:48)(cid:48) hj ) = T ( s (cid:48) hj s hj ) = T ( s (cid:48) hj ) T ( s hj ) = s (cid:48) thj s thj (55)Note that in spite (48), if prescription (50) is fulfilled,then s (cid:48) hj , s hj will commute at sector level (yet, prescrip-tion should be fulfilled for ± α in order that U (cid:48) h ( t ) , U h ( t )commute). Nevertheless, is easy note that (50) automat-ically implies j (cid:48) h − α = S α j h − α , b (cid:48) h − α = S α b h − α , S α = ± C h contains all sets of single commutative (not be-tween sets) evolution matrices of form U h ( t ) in (17).In the further discussion, the following additional setsand subgroups are notable in terms of group theory,which will be shown in Figure 2. The first one are thesubgroups in S ∗ h {| j h ∓ α |}{ s bh ∓ α } : S ∗ h {| j h ∓ α |}{ s bh ∓ α } , ≡ { U h ( t ) ∈ S ∗ h {| j h ∓ α |}{ s bh ∓ α } | ∆ h + α = 2 nπ, n ∈ Z } (56)which clearly correspond to those where sectors phases e ± i ∆ h + α have been removed. This subgroup contains the s hj, matrices defined in (22). In particular, S ∗ h { , }{ , } , is the diagonal matrix whose sectors have only oppositephases in each entry (taking s b h ∓ α = 0 as exception toour notation in (36) because | j h ± α | = 1 → | b h ± α | = 0).Matrices in this subgroup are responsible to introducerelative phases between Bell states associated with thesame sector. Other relevant subgroup is: D h ≡ { U h ( t ) ∈ S ∗ h {| j h ∓ α |}{ s bh ∓ α } | ∆ h −± α = n ± α π, n ± α ∈ Z } (57)they correspond precisely to matrices D φh presented be-fore, responsible to introduce relative phases betweenBell states associated with different sectors. These ma-trices are contained in any S ∗ h {| j h ∓ α |}{ s bh ∓ α } so they commutewith any matrix in C h and inclusively in S h . Thus, interms of group theory, D h is a normal group [51] of S h : D h (cid:67) S h . In a related sense, given a subgroup A in G ,the left cosets (similarly for right cosets) are defined assets: gA ≡ { ga | g ∈ G, a ∈ A } (58)0with this, we can set a result of group theory. Because D h is a normal group, then the set denoted by G/ D h of left(or right) cosets on bigger group G (possible groups rep-resented by G are S ∗ h { , }{ , } , , S ∗ h {| j h ∓ α |}{ s bh ∓ α } , S h or inclusively SU (4)) which contains D h , is the quotient group [51] of FIG. 2: Schematic representation of each S h for h = 1 , , SU (4). a) Structure of S h with an infinite number ofsections, each one containing a different subgroup S ∗ h {| j h ∓ α |}{ s bh ∓ α } parametrized in [0 , × ×{±} × and covering C h (dotted ovalregion). Sections extends on oval white region containing D h (indicated with the inner dotted boundary) in which all sub-groups S ∗ h {| j h ∓ α |}{ s bh ∓ α } share elements. Note specially the subgroup S ∗ h { , }{ , } and their subgroup S ∗ h { , }{ , } , . C h does not cover S h .Uncovered region (semi lunar region in white) could be di-vided in two regions by the gray circular dotted line: innerregion is the group (cid:104)C h (cid:105) , which contains all finite products ofelements in C h ; and outer one contains the elements in S h notachievable by any finite product of elements in C h . b) Combi-nation of three subgroups S h being contained in SU (4), whichshare elements in the central circular region E . G by D h . It can be understood as a homomorphism on G and it is easy note that D h is the kernel of this ho-momorphism (the set of elements in G which maps on D h ).If A, B are subgroups in G , then we define their prod-uct [51] as: AB = { ab | a ∈ A, b ∈ B } (59)which is not necessarily a group at least one of them bea normal group which is a result well known in grouptheory [51]. Then should be clear that: S ∗ h {| j h ∓ α |}{ s bh ∓ α } = D h S ∗ h {| j h ∓ α |}{ s bh ∓ α } , = S ∗ h {| j h ∓ α |}{ s bh ∓ α } , D h (60)Under this circumstances, this set represents how D h can be understood as a merely factor on the whole struc-ture of group G . Summarizing, in group theory language, S ∗ h {| j h ∓ α |}{ s bh ∓ α } = D h × S ∗ h {| j h ∓ α |}{ s bh ∓ α } , is a direct product of thesesubgroups [51], which physically means that evolutionoperators obtained with one magnetic pulse are equiv-alent to two pulses, one to provide sector phases andother to reproduce the remaining dynamics free of sectorphases.Finally, the subgroup: E ≡ { e iφ I | φ ∈ [0 , π ) } ⊂ (cid:92) h =1 , , S h (61)can be understood as responsible to introduce globalphases in quantum states under interaction (1) as dif-ference with those phases introduced by S ∗ h { , }{ , } , and D h . Clearly, as S ∗ h {| j h ∓ α |}{ s bh ∓ α } , and D h , E is isomorphicto U (1). Thus, similarly with D h , G/ E is a quotientgroup, representing how a global phase is a factor group U (1) in the structure of each possible G in this analysis( S ∗ h { , }{ , } , , S ∗ h {| j h ∓ α |}{ s bh ∓ α } , S h or SU (4)).Figure 2 depicts schematically the relations betweenbefore sets and subgroups in SU (4). Despite their rep-resentations (17), each S h (Figure 2a) has same groupstructure as a result of their isomorphic relations via ro-tations R , ( α, β, γ ). Inside, there are an infinite numberof subgroups S ∗ h {| j h ∓ α |}{ s bh ∓ α } parametrized on [0 , × ×{±} × ,which cover C h . Subgroup D h contains the common el-ements of them. Subgroups S ∗ h {| j h ∓ α |}{ s bh ∓ α } , (not indicatedin figure) are each one in their corresponding S ∗ h {| j h ∓ α |}{ s bh ∓ α } .When three groups S h ⊂ SU (4) are combined (Figure2b), subgroup E contains their only common elements.Thus, oval regions with dotted boundary, C h , in each S h represent the evolutions achievable with only one pulseof magnetic field. Instead, white semi lunar regions1Π C h = S h (cid:31) C h are complementary evolutions which acts’locally’ and unitarily in each matrix sector but not ableto be generated in just one pulse.Other important result in group theory is that givena set of groups, by example C h ∈ S h , then their gener-ated subgroup is defined as all finite products of elementsand/or their inverses: (cid:104)C h (cid:105) ≡ { u = a a . . . a n | n ∈ Z + , a i ∈ C h ∪ C − h } (62)where C − h ≡ { a − i | a i ∈ C h } (in this case C h = C − h ).Then, (cid:104)C h (cid:105) have the property of being a group, whichis easily demonstrable departing from group conditions.It means that finite products of elements in C h extendstheir coverage into S h forming a group [51], but it is notnecessarily whole S h (this group is depicted in Figure 2with a gray dotted circular line): C h ⊂ (cid:104)C h (cid:105) ⊂ S h . Notethat (cid:104)C h (cid:105) is generated by the minimum set: F h ≡ D h ∪ ( (cid:91) | j h ± α |∈ [0 , s bh ∓ α = ± S ∗ h {| j h ∓ α |}{ s bh ∓ α } , ) (63)thus, in terms of group theory, (cid:104)C h (cid:105) is a free group on F h .In these terms, Π C h (cid:31) (cid:104)C h (cid:105) contains all possible elementsin S h which are not achievable by any finite combinationof interaction pulses (1) for a fixed h . Elements in thislast set have the following property: p i p j = s k , ∀ p i , p j ∈ Π C h (cid:31) (cid:104)C h (cid:105) with s k ∈ (cid:104)C h (cid:105) . It follows that a) (cid:104)C h (cid:105) self-contains their inverses; b) for any cosets generated by p ∈ Π C h (cid:31) (cid:104)C h (cid:105) , then p ∈ p (cid:104)C h (cid:105) ; and c) by hypothesis,no one element of Π C h (cid:31) (cid:104)C h (cid:105) , can be obtained as a finiteproduct of elements of (cid:104)C h (cid:105) . As a corollary, the productof an even number of elements p i of Π C h (cid:31) (cid:104)C h (cid:105) becomeselement of (cid:104)C h (cid:105) , while the product of an odd numberremains in this set.Note that formula (52) is useful and easy to determinehow two pulses s (cid:48) hj , s hj with n (cid:48) , n not parallel generateelements outside of C h . Because form of n in (52), itshows too that for s haj , s hbj with given j ha ∓ α , j hb ∓ α and s ab h ∓ α , s bb h ∓ α respectively, then it is not possible to findcorrespondingly a s hcj which fulfill s haj s hbj = s hcj at least n a , n b will be parallel and then n b (owning to same sub-group). Then, finite products in C h are not trivial be-cause they can not easily be rearranged to simplify theirstructure, except in the case when two pulses belong tothe same group S ∗ h {| j h ∓ α |}{ s bh ∓ α } .As SU (2), each set of sectors in a given S ∗ h {| j h ∓ α |}{ s bh ∓ α } group is a Lie group with ∆ h − α as parameter. A result inLie group theory is that every element of the connectedsubgroup of any linear Lie group can be expressed as afinite product of exponentials of its real linear Lie algebra[51, 52]. So, if sectors of elements in (cid:104)C h (cid:105) are connected,it will imply that this last group is SU (2) really. We cananalyze connectivity with help of formula (52). Clearly elements in S ∗ h {| j h ∓ α |}{ s bh ∓ α } have n restricted to planes 1 − − x − z , y − z but not associated with physicaldirections) depending of h parity. Thus, n × n (cid:48) in (52)is orthogonal to these vectors. With this, we can definethe following orthonormal vector basis (Figure 3): nn ⊥ = n (cid:48) − cos δ n (64) n (cid:118) = csc δ n × n (cid:48) where cos δ = n · n (cid:48) . Then, it is possible to express n (cid:48)(cid:48) in(52) in a spherical system of coordinates as: n (cid:48)(cid:48) = cos α n + sin α cos β n ⊥ + sin α sin β n (cid:118) (65)becoming when we solve:∆ h −± α = β + nπ, n ∈ Z sin ∆ (cid:48) h −± α = sin ∆ (cid:48)(cid:48) h −± α sin α csc δ (66)cot δ = sin ∆ (cid:48)(cid:48) h −± α cos α cos β + cos ∆ (cid:48)(cid:48) h −± α sin β sin ∆ (cid:48)(cid:48) h −± α sin α which solves the desired output element in terms of inputparameters. Right side expression for sin ∆ (cid:48) h −± α rangesbetween [ − ,
1] as is required (it can be shown by com-bining two last expressions and then obtaining their ex-treme values). A brief analysis shows that cot δ rangesin R . Then, two pulses in different groups S ∗ h {| j h ∓ α |}{ s bh ∓ α } canbe adequately selected to reproduce a general element in SU (2) (still, cases when θ, α = nπ, n ∈ Z can be ob-tained as limit cases; some of them are cases discussed FIG. 3: Generation of an arbitrary element of SU (2) for sectorin elements of U h ( t ) with two pulses. Basis n , n ⊥ , n (cid:118) letexpress general components of n (cid:48)(cid:48) , letting to obtain solutionsfor it and ∆ (cid:48)(cid:48) h −± α in terms of ∆ h −± α , ∆ (cid:48) h −± α , δ . δ , theangle between original vectors n and n (cid:48) , more than repre-sentation of specific vectors being considered. With this,Π C h (cid:31) (cid:104)C h (cid:105) = ∅ . This result is useful because states thatelements in SU (2) are products of sectors in U h ( t ). Withthis, each element in S h can be obtained in a finite num-ber of pulses, each one belonging to S ∗ h {| j h ∓ α |}{ s bh ∓ α } groups.This analysis state algebraic properties for solutions(17) of Ising interaction model (1) being considered. Thisperspective lets combine them for different purposes incontrol, gates and states design between others. V. CONCLUSIONS
Physical systems as molecules to set databases [53],magnetic molecular clusters and dielectric nanometer-size single domain to set quantum information processing[54], spins in quantum dots formed in GaAs heterostruc-tures, nanowire-based quantum dots or self-assembledquantum dots as suitable qubits [55, 56], are some ex-amples of physical systems on which spin control hasbeen experimented. Part of idea in those systems is tohave sufficient ability to have single resources on whichset quantum computation and quantum information pro-cessing in terms of before quantum computer models [53]with programmable spinspin couplings as some of shownhere. It is clear through these examples that differentphysical systems can converge on very similar kind of in-teractions which require deep analysis of their models tobe experimentally exploited. In addition, it is clear thatas experimental and technology advance, then more finecontrol has been applied to control paths, stable equilib-rium, confinement and quantum states of course. Then,models with extended parameters of control should beanalyzed because them could bring a better performancein the quantum states control arena. These theoreticaldevelopments sooner or later meet with experimental de-velopments in order to become useful in quantum engi-neering.Nevertheless that models with several couplings includ-ing more that two qubits at time, these trends are studiedto improve some proposals of superdense coding, multi-entangled processing, quantum walks and other whichrequire it. Despite control is being improved, it has beenclear that decoherence in a multiqubits deployments in-creases easily with their parts number, so manipulatelots of qubits coordinately becomes difficult in a grow-ing array [54]. Thus, alternative well controlled devel-opments based on a few quantum qubits at time shouldbe developed to implement quantum algorithms beingconstructed specifically to these kind of systems whilemultipartite control is better understood and improved.Circuit-gate model of quantum computers could be basedin great extent on bipartite systems when ancilla qubitsare used. Clearly, extensions to a few more qubits will be neces-sary still because such quantum stuff requires a systemof several qubits to make some task efficiently and themain materials based technology known for that is mag-netic. The most of them exploits Ising interactions withdifferent approaches [53], together with control on quan-tum states and in particular with entanglement control,a milestone in all almost these researches. Analysis pre-sented in this work states some properties which gener-alize some restricted models used in several approachesand experimental setups. In this sense, three dimensionalmodel can reduce to simplified models but brings poten-tially extensions in those models and technologies.Circuit-gate model was the first approach to quantumcomputation, nevertheless, quantum annealing [57] ormeasurement-based quantum computation [58] are alter-natives which use magnetic systems approached by Isinginteractions to manage a planned and controlled quan-tum state manipulation. By example, [59] has proposeda scheme to simulate the Ising model and preserve themaximum entangled states (Bell states) in cavity quan-tum electrodynamics (QED) driven by a classical fieldwith large detuning. On them, several applied problemshas been exhibited as the goal of (these technologies (pat-tern matching, folding proteins, an other particular NP-complete problems [53]).In these directions, solution for Ising model presentedhere (1) can be applied for more controlled situations in-cluding more than three freedom degrees. Magneto-optictraps an QED cavities could tentatively manage controlof position and contain particles, ions or molecules leav-ing still three directions for spin-spin coupling. Still, notall of specific effects need use three dimensional freedomdegrees but other developments can extent their externaland internal dynamics into three dimensions (by exam-ple, [60], has been reported three dimensional tracking ofquantum dots).Future work for model and solutions (17) presentedcould be based in different research lines. One proposalis to grown the analysis to control of elementary pieces toset adequate resources in circuit-gate quantum computerin terms of (25-26). Control procedures as EvolutionLoops or Exchange Operations states a basic structureand language of manipulation to maintain or transformqubits selectively. This basic language lets translate thecircuit-gate quantum computation algorithms into physi-cal operations based on realistic systems. After of controlanalysis to set stable and recoverable quantum resources,other possible extensions are based on statement of acomputer grammar based on those resources (as statesFigure 1 and structure depicted in terms of Bloch sphereto pairs of Bell states. An outstanding aspect here wasthe introduction of non local basis to depicts dynamics,which uncover the regular forms in (17) with well un-derstood group structures. It is possible that for modelsincluding more qubits, this structure could be maintainedin terms of an adequate basis of maximal entangled statesas in (16) for two qubits. In terms of these expressions,3our interaction appear as operate almost independentlyon pairs of maximal entangled states.Another extension is to exploit possibilities for twopulses expression (39) or in general a finite product ofpulses, which extend group dominion on S h . Still, com-bination of those operators for different values of h shouldbe studied to state its coverage on SU (4).In this line of research, the analysis of behavior withfinite temperature based on matrix density is compulsoryto consider decoherence effects. At same time, error cor-rection analysis is necessary in procedures which emergeof present model, based on error factors (as magneticfield, knowledge and control of interaction strengths,time, etc.). In our approach, of course improvementsshould be generated through to alternative continuouspulses. Rectangular pulses are easy to manage theoreti- cally but reality is that they are experimentally few prac-tical because their discontinuity and associated resonanteffects.Just as control development advances, more complexmodels can be experimented. Nuclear magnetic reso-nance, Quantum dots and Electrons in silicon latticeshave been the most successful systems in implementingquantum algorithms based on their coherence and stabil-ity. 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